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For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots. [ABSTRACT FROM AUTHOR]

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new. [ABSTRACT FROM AUTHOR]

In this paper, we give two generalized homogeneousq-operator identities, which are obtained byq-difference equations. In addition, we generalize certain generating functions from Chen et al. [W.Y.C. Chen, A. M. Fu, and B. Zhang,The homogeneousq-difference operator, Adv. Appl. Math. 31 (2003), pp. 659–668] and Saad and Sukhi [H. L. Saad and A.A. Sukhi,Another homogeneousq-difference operator, Appl. Math. Comput. 215 (2010), pp. 4332–4339] by the method ofq-difference equations. Moreover, we gainU(n+1) generalizations of generating functions for Verma–Jain polynomials byq-difference equations. Finally, we obtain two transformational identities byq-difference equations. [ABSTRACT FROM AUTHOR]

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855). [ABSTRACT FROM AUTHOR]

In this paper, we consider kth order difference equations of the form[image omitted] where every function [image omitted] is continuous and monotonic in each of its arguments. We generalize and improve upon some previous techniques to prove convergence using invariant intervals as the common theme. We apply our techniques to several different special cases of the equation above. [ABSTRACT FROM AUTHOR]

The paper presents theorems on the global asymptotic behaviour of non-autonomous population models which for the special case of periodically forced populations generalize known results. It is shown that a non-autonomous discrete dynamical system on an interval possesses the property of path stability, if the system is given by cave functions. From this it follows for a periodic system with strictly cave functions that there exists a globally asymptotically stable cycle. Old and new examples are discussed. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach modules over a [image omitted] -algebra and let [image omitted] be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital [image omitted] -algebra:[image omitted] We show that if [image omitted] and an odd mapping [image omitted] satisfies the functional equation (0.1) then the odd mapping [image omitted] is Cauchy additive. As an application, we show that every almost linear bijection [image omitted] of a unital [image omitted] -algebra A onto a unital [image omitted] -algebra B is a [image omitted] -algebra isomorphism when [image omitted] for all unitaries [image omitted] , all [image omitted] , and all [image omitted] . The concept of generalized Hyers-Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [ABSTRACT FROM AUTHOR]

The Goulden-Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2]. In this paper, we elaborate on one of the several extensions in [2], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2]. The extended result is then compared with the method of Régnier-Szpankowski [3], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier-Szpankowski method can be simplified to the extended Goulden-Jackson method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear difference equation of the form xn+1 =f(pn,xn-2s,xn-2t-1), n = 0,1,…, under some certain assumptions, where s,t ∈ {0,1,2,…} with s ⩽ t and the initial values x-2t-1 ,x-2t,ߪ, x0 ∈ (0, + ∞) and pn is the period-two sequence. We give sufficient conditions under which every positive solution of this equation converges to the period-two solution. [ABSTRACT FROM AUTHOR]

In this paper, we study global attractivity of solutions of the delay difference equation with a forcing term where {p(n)} is a nonnegative sequence, f:R?R is a continuous function with xf(x)>0 for x?0, {r(n)} is a real sequence, and k is a nonnegative integer. Several global attractivity results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear difference equation of the form xn+1 =f(pn,xn-2s,xn-2t-1), n = 0,1,…, under some certain assumptions, where s,t ∈ {0,1,2,…} with s ⩽ t and the initial values x-2t-1 ,x-2t,ߪ, x0 ∈ (0, + ∞) and pn is the period-two sequence. We give sufficient conditions under which every positive solution of this equation converges to the period-two solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behaviour and the stability of a class of linear neutral delay difference equations with variable coefficients and constant delays. Via an appropriate solution of the so-called generalized characteristic equation, an asymptotic criterion and a stability result are established. [ABSTRACT FROM AUTHOR]

We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

We consider general, higher order difference equations of typein which the function f is non-increasing in each coordinate. We obtain sufficient conditions for the asymptotic stability of a unique fixed point relative to an invariant interval. We also discuss various applications of our main results. [ABSTRACT FROM AUTHOR]

The two-parameter absolute difference equation xn+1 = ∣axn-bxn-1∣ is studied. Based on the parameter values a, b and a pair of initial values, we consider the existence and bifurcations of solutions having one or more of the following properties: (i) unbounded, (ii) convergent (to zero or to a positive constant) (iii) monotonic, (iv) periodic and (v) non-periodic oscillatory. The semiconjugate first order equation satisfied by the ratios (xn/xn-1 is used to significant advantage for points (a, b) in certain regions of the parameter plane. Some open problems and conjectures are presented. [ABSTRACT FROM AUTHOR]

We derive and factorize the fourth-order q -difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q -classical orthogonal polynomials. These perturbations include the r th associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q -classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q -difference equations and express the modified families in terms of the starting ones. [ABSTRACT FROM AUTHOR]

The article discusss about some open problems and conjectures about some interesting types of difference equations.